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Many of the known functions are continuous in open interval. Polynomial, trigonometric, exponential, logarithmic functions etc. are continuous functions in open interval.

Continuity in a closed interval [a,b]

The possibility that there always exist a point around a given point is not there at end points of closed interval. We can not determine left limit at lower end and right limit at upper end of the closed interval. For this reason, we test continuity of function at the closing points from one side only. For a function to be continuous in the closed interval, it should be continuous at all points in the interval and also at the bounding values of closed intereval, [a,b]. Hence,

(i) limit exists at all points in the interval and are equal to function values at those points.

lim x > c f x = f c ; a < c < b

(ii) right limit exists at x=a and is equal to function value at the lower end of closed interval. .

lim x > a + f x = f a

(iii) left limit exists at x=b and is equal to function value at the upper end of closed interval.

lim x > b f x = f a

Function operations, compositions of function and continuity

If two functions are continuous at a point or in interval, then function resulting from function operations like addition, subtraction, scalar product, product and quotient are continuous at that point. Further, properly formed function compositions of two or more functions are also continuous.

These properties of continuity are extremely helpful tool for determining continuity of more complicated functions, which are formed from basic functions. Idea is that we are aware of continuity of basic functions. Therefore, continuity of functions formed from these basic functions will also be continuous.

Generally, basic functions are continuous in real numbers set R or its subsets. For example, we know that polynomial functions, sine, cosine, tangent, exponential and logarithmic functions etc are continuous on R. Similarly, a radical function is continuous for non-negative x values. Their composition or the new function will be continuous in the new domain, which is defined in accordance with the rule given here :

  • scalar product (multiplication with a constant) : domain remains same
  • addition/subtraction/product : domain is intersection of individual domains
  • division or quotient : domain is intersection of individual domains minus points for which denominator is zero
  • fog or gof : domain is intersection of individual domains

In the nutshell, the function formed from other functions is continuous in new domain as defined above. If we look closely at the definition of continuity here, then "finding interval in which function is continuous" is same as finding "domain" of new function arising from mathematical operations.

Continuous extension of function

Many functions are not defined at singularities. For example, rational functions are not defined for values of x when denominator becomes zero. By including these singular points or exception points in the domain, we can redefine function such that it becomes continuous in the extended domain. This extension of the domain of function such that function remains continuous is known as continuous extension.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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